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Volatility SmilePeter Nowak and Patrick Sibetz† covering SABR ModelHeston Model† coveringApril 24, 2012AbstractThis document is analysing two famous stochastic volatily models, namelySABR and Heston. It introduces the problems of the Black Scholes model,the two stochastic models and in a final step calibrates volatilty smiles/surfaces for given FX option data.Key words: smiles, skew, implied volatilities, stochastic volatilities, SABR, Heston1

Contents1 Introduction32 Black’s model with implied volatilities42.1FX Black Scholes Framework . . . . . . . . . . . . . . . . . . . .42.2FX Implied Volatility Smile . . . . . . . . . . . . . . . . . . . . .43 Stochastic volatility models3.13.23.37The Heston model . . . . . . . . . . . . . . . . . . . . . . . . . .73.1.1Heston Option Price . . . . . . . . . . . . . . . . . . . . .103.1.2Characteristic Functions . . . . . . . . . . . . . . . . . . .123.1.3Heston FX Option Extension . . . . . . . . . . . . . . . .153.1.4Summary for Heston Option Pricing . . . . . . . . . . . .17The SABR model . . . . . . . . . . . . . . . . . . . . . . . . . . .193.2.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . .193.2.2SABR Implied Volatility . . . . . . . . . . . . . . . . . . .203.2.3Model dynamics . . . . . . . . . . . . . . . . . . . . . . .213.2.43.2.5Parameter Estimation . . . . . . . . . . . . . . . . . . . .SABR Refinements . . . . . . . . . . . . . . . . . . . . . .2224Calibration and Simulation . . . . . . . . . . . . . . . . . . . . .263.3.1The Heston Model . . . . . . . . . . . . . . . . . . . . . .263.3.2The SABR model30. . . . . . . . . . . . . . . . . . . . . .4 Conclusio375 Appendix395.15.2Heston Riccati differential equation . . . . . . . . . . . . . . . . .Analysis of the the SABR model . . . . . . . . . . . . . . . . . .39405.3Implementation in R . . . . . . . . . . . . . . . . . . . . . . . . .465.3.1FX Black Scholes Framework . . . . . . . . . . . . . . . .465.3.2Heston Framework . . . . . . . . . . . . . . . . . . . . . .475.3.3SABR Framework . . . . . . . . . . . . . . . . . . . . . .482

1IntroductionAccording to the Black-Scholes model, we should expect options that expireon the same date to have the same implied volatility regardless of the strikes.Thus, Black erroneously assumed that the volatility of the underlying is constant. However, implied volatilities vary among the different strike prices. Thisdiscrepancy is known as the volatility skew or smile. In general, at-the-moneyoptions tend to have lower volatilities that in- or out-of-the-money options, seefigure 1.For estimating and fitting such volatility smiles, in terms to accuaratly price options, several frameworks have been introduced. Merton [10] suggested to makethe volatility a deterministic function of time. This would indeed explain thedifferent volatility for different tenors, but would not explain the smile effect fordifferent strikes. Other local volatility models introduced by Dupire [4], or theone from Derman and Kani [3], including a state dependent volatility coefficientyields still a complete market model, but it cannot explain the persistent smileshape which does not vanish over time with longer maturities.Thus the next step would be to allow the volatility to move stochastically overtime, where we will chose models to fit a certain market implied volatility surface. This processes were pioneered by Hull and White [8], Heston [7] and laterby Patrick Hagan through the widely used SABR model [6].3

2Black’s model with implied volatilitiesThe implied volatility of a european option in the Black Scholes framework is analternative of quoting the options price, as every other parameter are observablein the market. If the market price of the option is quoted one find σimp thatthe Black Scholes option model price C BS equals the option’s market price.C BS (S, K, σimp , rd , rf , t, T ) C (1)As the option price is monotonically increasing in the volatility it can be uniquelydetermined. However, as the BSM formula cannot be solved for the volatilityanalytically, a numerical algorithm has to optimise this approach.For our examples we will analyse the FX option implied volatility surface, ascurrencies tend to provide the so called volatility smile in general. Therefore weneed also to introduce the differences to the normal Black Scholes framework.2.1FX Black Scholes FrameworkFor the FX smile we will consider a model for the FX spot rates to be strictlypositive and evolve stochastically over time. In our model framework we willadapt the Black Scholes model [1] with the model of Garman and Kohlhagen[5] which is also based onto Black Scholes as an application to foreign currencyoptions. The following analysis is based on risk neutral valuation, which meansthat a risk free portfolio will always yield the risk free interest rate in the respective currency. Otherwise there would be an arbitrage opportunity leadingto a risk free profit.The important point of risk neutral valuation is, that the underlying asset itselfis a risky investment and therefore also derivatives based on it are risky too.However, it is possible to costruct an instantaneously risk free portfolio consisting of the two securities. The proportions of the two securities however, are notstatic, but need constantly be adapted over time. This process is thus oftenreferred to as dynamic hedging.2.2FX Implied Volatility SmilePerhaps due to that American options are almost not traded in FX markets,the market uses the Black Scholes formula for price quotation. These quotesare in Black Scholes implied volatilities rather than the prices directly. What isalso a peculiarity of the FX market is that quotes are provided at a fixed Black4

Scholes delta not at fixed strike levels as it is usual for options in other markets.In particular the options are quoted implicitly for five different levels of deltafor different tenor points. These standard moneyness levels are at the moneylevel, 25 delta out of the money level and 25 delta in the money level (75 delta)and the same for 10 delta.FX Volatility Smile Implied VolatilityRR10 BF10 ATM10C25CATM25P10PDeltaFigure 1: FX Smile including the three point market convention quotationSince out of the money levels are liquid moneyness levels in the options market,market quotes these levels as 25 delta call and 25 delta put. If a trader hasthe right model, he can build the whole volatility smile for any time to expiryby using the three points in the volatility surface. The additional two points of10 delta options yields a better calibration as far out of the money options canhave even higher than extrapolated implied volatility. In the options market 25delta call and 25 delta put points are not quoted as volatility. They are quotedaccording to their positions to at the money volatilty level. These parametersare 25 delta butterfly and 25 delta risk reversal.Risk Reversal:Risk reversal is the difference between the volatility of the call price and theput price with the same moneyness levels. 25 delta risk reversal is the differencebetween the volatility of 25 delta out of the money Call and 25 delta out of themoney Put.5

RR25 σ25C σ25PButterfly:Butterfly is the difference between the avarage volatility of the call price and putprice with the same moneyness level and at the money volatility level. In otherwords for example for 25 delta level, butterfly defines how far the average volatility of 25 delta call and 25 delta put is away from the at the money volatiltiy level.BF25 (σ25C σ25P )/2 σAT MA real world example shall motivate the necessity to apply option pricing models that are richer than the classical model of Black and Scholes (1973, [1]). Itshows that the Black-Scholes implied volatilities for EUR/JPY FX options fordifferent deltas and maturities.Figure 2: Bloomberg market data for the USDJPY implied volatility surfaceThe volatility surface of this dataset then looks the following6

0.130.120.110.090.10Implied Volatility0.140.15USDJPY FX Option Volatility Smile10C25CATM25P10PDeltaFigure 3: USDJPY implied volatility surface3Stochastic volatility models3.1The Heston modelA well established model to price equity options including a volatility smile orskew in practice is the Heston Model. Here the underlying follows a diffusionstochastic process, like in the Black Scholes model, but the process’ stochasticvariance ν follows a Cox Ingersoll Ross (CIR) process. dSt µSt dt dνt νt St dWtS κ(θ νt )dt σ νt dWtνdWtS dWtν ρdtIn this model the positive volatility of the underlyings volatility σ generates asmile, and a nonzero correlation ρ generates a skew of the volatility curve withslope of the same sign.The parameters in this model are: µ the drift of the underlying process κ the speed of mean reversion for the variance θ the long term mean level for the variance7

σ the volatility of the variance ν0 the initial variance at t 0 ρ the correlation between the two Brownian motionsTo derive the semianalytic solution for the FX Option we will first begin withthe classical Heston model for equity options and then show the extension tothe two currency world model. We begin with a portfolio consisting of one assetmore than in the Black Scholes replication approach, as we have also one moreBrownian motion driving the underlying’s volatility. Thus we have a portfolioconsisting of one option V (S, ν, t) a portion of the underlying St and a thirdderivative to hedge the volatility φU (S, ν, t). The portfolio has then the valueΠt V (S, ν, t) St φU (S, ν, t).The next assumption we make is that the portfolio is selffinancing which bringsus to the following equation which describes the change in value of the portfolio:dΠ dV dS φdUFor the two Options U and V we apply the Ito-formula to expand dU (S, ν, t):11dU Ut dt US dS Uν dν USS (dS)2 USν (dSdν) Uνν (dν)222With the quadratic variation and covariation terms expanded we get(dS)2(dSdν)(dν)2 d hSi νS 2 d W S νS 2 dt, d hS, νi νSσd W S , W ν νSσρdt, and d hνi σ 2 νd hW ν i σ 2 νdt.The other terms including d hti , d ht, W ν i , d t, W S are left out, as the quadraticvariation of a finite variation term is always zero and thus the terms vanish. ThusdU 11Ut dt US dS Uν dν USS νSdt USν νSσρdt Uνν σ 2 νdt22 112Ut USS νS USν νSσρ Uνν σ ν dt US dS Uν dν22 {z} :AUWe analogously define the term AU for the derivative V as AV . We then getthe two siplified equations for the developments of the two derivatives asdU AU dt US dS Uν dνdV AV dt VS dS Vν dν8

Thus the portfolio evolution is described by the following PDE dΠ φAU AV dt (VS φUS ) dS (Vν φUν ) dν {z} {z}!! 0 0including a constant hedge by adapting the positions φ and like in the BSMmodel that the stochastic terms dS and dν vanish. We herewith get a uniquesolution for the two positions with:φ VνUν VS φUS VS Vν USUνWe further know as we have introduced before in our risk neutral valuationassumptions, that under the risk neutral pricing measure the expected portfolioreturn must equal the risk free interest rate r.dΠ rΠdt r(V S φU )dtThus changes in the riskless protfolio are described by dΠ φAU AV dt φAU AV r(V S φU )dt r(V S φU )If we now plug in the solution for and φ derived above, we getVν AU AVUν Vν AU AV UνAV rV rSVSVν Vν USrV rS VS Uν rVνUUνrV Uν rSVS Uν rSVν US rVν UAU rU rSUSUνSo both sides only dependend on the respective derivative U or V , which weaccording to Heston assume to be representable by a deterministic functionf (t, ν, S)f (t, ν, S) κ(θ νt ) λ(t, ν, S)The term λ(t, ν, S) is called the market price of volatility risk. Heston assumedit to be linear in the instantaneous variance νt , i.e. λ(t, ν, S) λ0 νt , in orderto retain the form of the equation under the transformation from the statisticalmeasure to the risk-neutral measure.9

Resubstituting then yieldsAU rU rSUSUν κ(θ νt ) λ(t, ν, S)11Ut USS νS 2 USν νSσρ Uνν σ 2 ν rU rSUS22 κ(θ νt ) λ(t, ν, S) UνWe now have derived exactly the same equation as Heston in his paper.3.1.1Heston Option PriceTo derive the price of a option we begin by stating a general risk neutral pricingapproach where we assume the price to be the discounted expected future payoffunder the pricing measure. As we know the payoff of a European plain vanillacall option to beCT (ST K) we can generally write the price of the option to be at any time point t [0, T ]:Ct e r(T t) E (ST K) Ft e r(T t) E (ST K)1(ST K) Ft e r(T t) E ST 1(ST K) Ft e r(T t) KE 1(ST K) Ft {z} {z} :( ) :( )With constant interest rates the stochastic discount factor using the bank account Bt then becomes 1/Bt e Rt0rs ds e rt . To be able to calculate theprobabilities above in the first term, we now need to perform a Radon-Nikodymchange of measure.Zt St BTdQFt dPBt STThus the first term ( ) gets( ) e r(T t) EP ST 1(ST K) Ft Bt P E ST 1(ST K) FtBT Bt Q E Zt ST 1(ST K) FtBT Bt Q St BT EST 1(ST K) FtBTBt ST Q E St 1(ST K) Ft St EQ 1(ST K) Ft St Q (ST K Ft ) St P1 (St , νt , τ )10

The second term can be calculated analogously even without the change ofmeasure.( ) e r(T t) KE 1(ST K) Ft e r(T t) KP (ST K Ft ) e r(T t) KP2 (St , νt , τ )Thus we get a pricing formulaCt St P1 (St , νt , τ ) e r(T t) KP2 (St , νt , τ )(2)which is comparable to the Black Scholes equation, however the two probabilitiesare different as we have stochastic volatility and thus a different distribution asthe normally distributed returns in the Black Scholes world. So what we nowneed to determine is the probability distribution of S under the two measures.P (ST K) P (ln ST ln K) 1 P (ln ST ln K) 1 Fln S (K)For the second term P2 the evolution is under the real probability measure Pwhereas the dynamics of P1 are under the new measure Q which we also needto determine the new distribution. To get a nicer form of the PDE we thenperform a change of variable X : ln S, which means we also need to adapt ourPDE from above due to changes of the derivatives with respect to S:USUSSUSν UX1S11 UX 22SS1 UXνS UXXIf we plug this in our PDE the form gets much nicer as due to the substitutionof the variables all the S terms cancel and we get the Heston PDE for the logprice X ln S. 1 21νS USS ρσνSUSν σ 2 νUνν rSUS κ(θ νt ) λ0 νt Uν rU Ut 022 1 21111νS UXX 2 UX 2 ρσνSUXν σ 2 νUνν 2SSS2 1 rSUX κ(θ νt ) λ0 νt Uν rU Ut 0S 11 21νUXX ρσνUXν σ νUνν r ν UX κ(θ νt ) λ0 νt Uν rU Ut 022211

3.1.2Characteristic FunctionsThe next step to finally reach an option price formula is to get the two probabilities in the pricing formulas, which is being done by using the characterisitcfunctions. As we face two different probability measures we also need to findtwo characteristic functions of the two probabilities P1 and P2 : iuX QeFtϕ1 (u) : ϕQX (u St , νt , τ ) E ϕ2 (u) : ϕPX (u St , νt , τ ) EP eiuX FtHeston assumed the form of the characteristic functions to beϕi (u) eCi (τ,u) Di (τ,u)ν iux(3)which we of course apply also for our approach. To get now the probabilities andtheir cumulative distribution function needed, we apply the Fourier InversionFormula on the characteristic function1T 2πZTFX (x) FX (0) lim Teiux 1ϕX (u)du iuand apply the solution of Gil-Pelaez to get the nicer real valued solution of thetransformed characteristic function:P(X x) 1 FX (x) 11 2 πZ 0 e iuxϕX (u) duiuwhich is the seeked solution for P1 and P2 iux Z11 ePj ϕjX (u) du2 π 0iu(4)Note that the function Pj is a cumulative distribution function (in the variablex ln K) of the log-spot price. after time τ : T t starting at X ln St forsome drift µ.P2 P (ln ST ln K Ft )The same applies for P1 under the risk neutral measure Q. However, it onlydoes depend on the last value of St and vt and is thus Markov which is quiteuseful as a property. To find now the two characteristic functions and theirfactors which generate them, we make use of the pricing formula (2), where wedetermine the derivatives with respect to t, X and ν and plug it into the PDE12

of the log price. C Ct t C CX XCXXCXνCνCνν eX Pt1 Kre r(T t) P 2 Ke r(T t) Pt2 eX Pt1 Ke r(T t) rP 2 Pt212 e X P 1 e X PX e r(T t) KPX 12 e X P 1 PX Ke r(T t) PX1112 e X PX e X P 1 e X PX eX PXX e r(T t) KPXX 112 e r(T t) KPXX eX P 1 2PX PXX12 eX Pν1 eX PXν e r(T t) KPXν 12 eX Pν1 PXν e r(T t) KPXν eX Pν1 Ke r(T t) Pν212 eX Pνν Ke r(T t) PννAs we now have a dependence on τ rather than the current valuation date t weneed to substitute this term, which brings f τ f t t τ f τ , thus our PDElooks the following: 11 21νCXX ρσνCXν σ νCνν r ν CX κ(θ νt ) λ0 νt Cν rC Cτ 0222(5)Before we plug the entire terms from above in the equation we will differentiatebetween two special cases, as the pricing PDE is always fulfilled irrespective ofthe terms in the call contract. S 1, K 0, r 0 Ct P1 S 0, K 1, r 0 Ct P2Since P2 follows the PDE so does P2 , so we have got two simplifications for thepricing PDE which for we will now plug in the derivatives determined above. 1 2 11111ν P 1 2PX PXX ρσν Pν1 PXν σ νPνν 22 11 κ(θ νt ) λ0 νt Pν1 rP1 Pτ 0 r ν P 1 PX2If we now rearrange these terms we see that the P1 term cancels out: 1 11 2 1111νP ρσνPXν σ νPνν r ν PX ρσν κ(θ νt ) λ0 νt Pν1 Pτ1 02 XX22(6)Now we perform the same steps for the second scenario including S 0 andK 1 so that only P2 remains in 1 21 2 22νP ρσνPXν σ νPνν r 2 XX2the option price. 12ν PX κ(θ νt ) λ0 νt Pν2 rP 2 rP 2 Pτ2 0213

Also here the P2 term cancels out and the equation gets: 1 21 2 2122νPXX ρσνPXν σ νPνν r ν PX κ(θ νt ) λ0 νt Pν2 Pτ2 0 (7)222For convenience we define a new variable ζ i which has the two values ζ 1 1and ζ 2 1, thus we can write a general formula for the two PDEs (6) and (7). 1 i1 2 iζ1 ζiiνP ρσνPXν σ νPνν r ν PX ρσν κ(θ νt ) λ0 νt Pνi Pτi 02 XX222(8)Applying the results of the Feynman-Kac theorem, the characteristic functionwill then also follow the Heston PDE. 1ζ1 ζ1 iνϕXX ρσνϕiXν σ 2 νϕiνν r ν ϕiX ρσν κ(θ νt ) λ0 νt ϕiν ϕiτ 02222(9)The respective derivatives we determine by the given structure of the characteristic function in (3). ϕi τ ϕi X 2 ϕi X 2 ϕi ν 2 ϕi ν 2 2 ϕi X ν Ci Di ν ϕi τ τ iuϕi u2 ϕi Di ϕi Di2 ϕi iuDi ϕiSo if we plug in the derivatives of the assumed structure of the characteristicfunction the PDE gets 11ζ νu2 ϕi ρσνiuDi ϕi σ 2 νDi2 ϕi r ν iuϕi 222 1 ζ Ci Di ρσν κ(θ νt ) λ0 νt Di ϕi ν ϕi 02 τ τ(10)The derivation of the solution for the PDE can be looked up in the appendix(5.1). The solution for Di looks as follows:(b di ) 1 edi τDi σ 2 (1 gi edi τ )wheregi b dib di14

anddi pb2 σ 2 (ζiu u2 )The second equation can then easily be determined by simply integrating out Ci τCi riu κθDi 1 gi edi τκθ riuτ 2 (b di )τ 2 lnσ1 giThus we have derived all components of the semi-analytic pricing equation forHeston options.3.1.3Heston FX Option ExtensionBefore we go now directly to the FX Heston model we will first derive theframework extension of Black Scholes for FX markets by Garman and Kohlhagen[5]. For the two currency framework we make use of the work of Musiela andRutkowski [12].FX Black Scholes ModelWe henceforth denote Btd and Btf for t [0, T ] as the price of the domestic andforeign saving account each denominated in units of their respective currency.The exchange rate process Qt is the price of units of domestic currency for 1 unitof the foreign currency and is described under the actual probability measure PbydQt µQt dt σQt dWtThe solu

This document is analysing two famous stochastic volatily models, namely SABR and Heston. It introduces the problems of the Black Scholes model, the two stochastic models and in a nal step calibrates volatilty smiles/-surfaces for given FX option data. Key words: smiles, skew, implied volatilities, stochastic volatilities, SABR, Heston 1

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